Peel and cut carrots into processor-sized pieces.

Cut red bell pepper into 1-inch-wide strips.

Fit food processor with shredding blade and put lid in place.

Shred carrots and peppers according to manufacturer's directions.

Place garlic cloves in feed tube and top with ginger.

Process garlic and ginger until finely chopped.

Place sugar, cider vinegar, water, salt, and cayenne pepper in a 2-quart saucepan.

Bring the mixture to a boil over medium-high heat.

Reduce heat to low, add the processed garlic and ginger, and cook for 3 minutes.

Add the shredded carrots and red bell pepper to the saucepan.

Bring the mixture back to a boil.

Reduce heat to low and cook for 3 minutes more, stirring occasionally.

Remove the saucepan from the heat and allow the relish to cool to room temperature.

Refrigerate the relish for at least 30 minutes to allow the flavors to meld.